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Robust exponential attractors for non-autonomous equations with memory
Feedback control via inertial manifolds for nonautonomous evolution equations
1. | Fachrichtung Mathematik, Technische Universitat Dresden, 01062 Dresden, Germany |
2. | Department of Mathematics, Dresden University of Technology, 01062 Dresden |
References:
[1] |
L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280. |
[2] |
L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925.
doi: doi:10.1016/S0362-546X(97)00569-5. |
[3] |
P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991). |
[4] |
S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.
doi: doi:10.1016/0022-0396(88)90007-1. |
[5] |
I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380.
doi: doi:10.1023/A:1016684108862. |
[6] |
C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180. |
[7] |
C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149.
doi: doi:10.1007/BF01762184. |
[8] |
D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992. |
[9] |
J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511.
doi: doi:10.1016/0022-247X(75)90193-6. |
[10] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353. |
[11] |
C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244.
doi: doi:10.1007/BF01047831. |
[12] |
C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288. |
[13] |
A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50. |
[14] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981. |
[15] |
N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941.
doi: doi:10.1023/A:1020768711975. |
[16] |
M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989. |
[17] |
Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268.
doi: doi:10.3934/dcds.1999.5.233. |
[18] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995. |
[19] |
L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987. |
[20] |
J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866. |
[21] |
A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47.
doi: doi:10.1070/IM1994v043n01ABEH001557. |
[22] |
R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86.
doi: doi:10.1023/A:1026153311546. |
[23] |
H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92.
doi: doi:10.1093/imamci/11.1.75. |
[24] |
H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42.
doi: doi:10.1006/jmaa.1995.1396. |
[25] |
G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255.
doi: doi:10.1016/0022-0396(92)90152-D. |
[26] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002. |
[27] |
S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184.
doi: doi:10.1016/S0959-1524(99)00029-3. |
[28] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997. |
show all references
References:
[1] |
L. Arnold and I. Chueshov, Order-preserving random dynamical systems: Equilibria, attractors, applications, Dyn. Stab. Syst., 13 (1998), 265-280. |
[2] |
L. Boutet de Monvel, I. D. Chueshov and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Analysis, 34 (1998), 907-925.
doi: doi:10.1016/S0362-546X(97)00569-5. |
[3] |
P. Brunovsky, Controlling the dynamics of scalar reaction diffusion equations by finite dimensional controllers, Proc. IFIP-IIASA Conf., Laxenburg/Austria 1989, Lect. Notes Control Inf. Sci. 154, 22-27 (1991). |
[4] |
S. N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.
doi: doi:10.1016/0022-0396(88)90007-1. |
[5] |
I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Differ. Equations, 13 (2001), 355-380.
doi: doi:10.1023/A:1016684108862. |
[6] |
C. M. Dafermos, An invariance principle for compact processes, J. Differ. Equations, 9 (1971), 239-252; Erratum. Ibid. 10, 179-180. |
[7] |
C. M. Dafermos, Semiflows associated with compact and uniform processes, Math. Systems Theory, 8 (1975), 142-149.
doi: doi:10.1007/BF01762184. |
[8] |
D. Daners and P. Koch Medina, "Abstract Evolution Equations, Periodic Problems and Application," Pitman Research Notes in Mathematics Series, vol. 27, Longman Scientific & Technical, Harlow, 1992. |
[9] |
J. Eisenfeld and V. Lakshmikantham, Comparison principle and nonlinear contractions in abstract spaces, J. Math. Anal. Appl., 49 (1975), 504-511.
doi: doi:10.1016/0022-247X(75)90193-6. |
[10] |
C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differ. Equations, 73 (1988), 309-353. |
[11] |
C. Foias, G. R. Sell and E. S. Titi, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, J. Dyn. Differ. Equations, 1 (1989), 199-244.
doi: doi:10.1007/BF01047831. |
[12] |
C. Foias, B. Nicolaenko, G. R. Sell and R. Temam, Varieties inertielles pour l'equation de Kuramoto-Sivashinsky. (Inertial manifolds for the Kuramoto-Sivashinsky equation), C. R. Acad. Sci., Paris, Ser. I, 301 (1985), 285-288. |
[13] |
A. Yu. Goritskiĭ and V. V. Chepyzhov, The dichotomy property of solutions of quasilinear equations in problems on inertial manifolds, Sb. Math., 196 (2005), 485-511; translation from Mat. Sb., 196 (2005), 23-50. |
[14] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, vol. 850, Springer, 1981. |
[15] |
N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dyn. Differ. Equations, 14 (2002), 889-941.
doi: doi:10.1023/A:1020768711975. |
[16] |
M. A. Krasnoselski, Je. A. Lifshits and A. V. Sobolev, "Positive Linear Systems, the Method of Positive Operators," Heldermann Verlag, Berlin, 1989. |
[17] |
Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete and Continuous Dynamical Systems, 5 (1999), 233-268.
doi: doi:10.3934/dcds.1999.5.233. |
[18] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Birkhäuser, Basel, 1995. |
[19] |
L. T. Magalhães, The spectrum of invariant sets for dissipative semiflows, Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, pp. 161-168, 1987. |
[20] |
J. Mallet-Paret and G. R. Sell, Inertial manifolds for reaction diffusion equations in higher space dimensions, J. Am. Math. Soc., 1 (1988), 804-866. |
[21] |
A. V. Romanov, Sharp estimates of the dimension of inertial manifolds for nonlinear parabolic equations, Russ. Acad. Sci., Izv., Math., 43 (1994), 31-47.
doi: doi:10.1070/IM1994v043n01ABEH001557. |
[22] |
R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee-Infante equation, J. Dyn. Differ. Equations, 15 (2003), 61-86.
doi: doi:10.1023/A:1026153311546. |
[23] |
H. Sano and N. Kunimatsu, Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems, IMA J. Math. Control Inform., 11 (1994), 75-92.
doi: doi:10.1093/imamci/11.1.75. |
[24] |
H. Sano and N. Kunimatsu, An application of inertial manifold theory to boundary stabilization of semilinear diffusion systems, J. Math. Anal. Appl., 196 (1995), 18-42.
doi: doi:10.1006/jmaa.1995.1396. |
[25] |
G. R. Sell and Y. You, Inertial manifolds: The non-self-adjoint case, J. Differ. Equations, 96 (1992), 203-255.
doi: doi:10.1016/0022-0396(92)90152-D. |
[26] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, vol. 143, Springer, New York, 2002. |
[27] |
S. Y. Shvartsman, C. Theodoropoulos, R. Rico-Martinez, I. G. Kevrekidis, E. S. Titi and T. J. Mountziaris, Order reduction for nonlinear dynamic models of distributed reacting systems, Journal of Process Control, 10 (2000), 177-184.
doi: doi:10.1016/S0959-1524(99)00029-3. |
[28] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," 2nd ed., Applied Mathematical Sciences, vol. 68, Springer, New York, 1997. |
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